Electrical control of the g-tensor of the first holein a silicon MOS quantum dot.

S. D. Liles1,†, F. Martins1, D. S. Miserev1,2, A. A. Kiselev3, I. D. Thorvaldson1,M. J. Rendell1, I. K. Jin1, F. E. Hudson4, M. Veldhorst4,5, K. M. Itoh6, O. P. Sushkov1,

T. D. Ladd1,3, A. S. Dzurak4, A. R. Hamilton1

1School of Physics, University of New South Wales, Sydney NSW 2052, Australia2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

3HRL Laboratories, LLC, 3011 Malibu Canyon Rd., Malibu, CA 90265, USA4School of Electrical Engineering and Telecommunications,

The University of New South Wales, Sydney NSW 2052, Australia5QuTech and Kavli Institute of Nanoscience, TU Delft, 2600 GA Delft, The Netherlands

6School of Fundamental Science and Technology, Keio University, Yokohama, Japan(Dated: December 10th, 2020)

AbstractSingle holes confined in semiconductor quantum dots are a promising platformfor spin qubit technology, due to the electrical tunability of the g-factor of holes.However, the underlying mechanisms that enable electric spin control remain uncleardue to the complexity of hole spin states. Here, we present an experimental andtheoretical study of the g-factor of a single hole confined in an isotopically enrichedsilicon planar MOS quantum dot. Electrical characterisation of the 3×3 g-tensorshows that local electric fields can tune the g-factor by 500%, and we observe a sweetspot where dg(110)/dV = 0, offering a configuration to suppress spin decoherencecaused by electrical noise. Numerical simulations show that non-uniform electrode-induced strain has an unexpectedly large influence on the coupling of hole spinsto electric fields in these spin-qubit devices. These results open a path towards apreviously unexplored technology; designer strain engineering to optimise hole spin-qubits.

Single hole spins confined in group IVquantum dots provide a promising path

towards scalable quantum computing1–4. Thesedevices can leverage well-established industrialplatforms5,6, while also enabling rapid all-electric spin control7–10. Recent demonstrationshave included single-qubit and two-qubit gateoperations of holes in Si and Ge devices10–15.

When developing spin qubit technology, afundamental question arises: What defines thecoupling of a single isolated spin to the externalmagnetic field? While this has been well studiedfor electrons2,16, the complexity of hole spinstates makes this a non-trivial question17–23.Holes occupy the valence band, which originatesfrom l=1 atomic p-orbitals, with an effectivespin of S = 3

2 and a strong intrinsic spin-orbitcoupling. For qubit devices, the combinationof spin-orbit coupling and quantum confinementstrongly modifies the hole spin properties, whichbecome sensitive to the size and shape of thequantum dot24,25. In addition, mixing betweenthe mj=±3/2 and mj=±1/2 hole sub-bandsleads to a mixed spin character, causing holespins to be sensitive to crystal anisotropies, localstrain effects, and the confinement profile21,26,27.

The g-tensor is the key parameter for studyingthe coupling of a spin-orbit state to a magneticfield27–34. However, most studies of the g-tensor of hole quantum dots have been performedusing devices that confine an unknown numberof holes35–42. This has hindered the abilityto understand hole spin-qubit devices, since thenumber of holes is a primary factor influencingthe orbital physics of the quantum dot43.It is imperative to know the quantum dotorbital wavefunction shape in order to make anyquantitative comparison between experimentsand theory, or to compare between differentdevice designs or material systems.

In addition, the sensitivity of the hole g-tensor to local electric fields (parameterised asd←→g /dV , for tensor ←→g and voltage source VG) isa key parameter for the maximum electricallydriven spin resonance (EDSR) frequency35,44,the potential for electric control of qubitaddressability, and the extent to which electricalnoise limits T ∗2 . Therefore, characterising theg-tensor of a single hole in a known andreproducible orbital configuration is a key stepfor progressing spin-qubit technology.

Here we study the spin properties of the† corresponding author - s.liles@unsw.edu.au

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Figure 1: A single hole in a silicon quantum dot. a, False colour SEM image of a device gate stack. Asingle quantum dot is formed in the region indicated by the white dashed circle by using gate G2 as the quantumdot plunger gate, while gates G1, G3 and the C-gate provide the electrostatic confinement. The in-plane crystalorientations are indicated, where the sample x-axis is the crystal axis [110], and the sample y-axis is the crystalaxis [110]. The out-of-plane direction is the sample z-axis, corresponding to crystal axis [001]. The horizontalwhite scale bar is 250 nm. b, Charge stability diagram showing operation down to the last hole, where thegray-scale is dIsens/dVG2. A series of charge transitions can be observed as negative spikes in dIsens/dVG2. c,The charging energy, EC , measured at the VG4 indicated by the colored horizontal arrows in a. Schematics onthe right indicate a line cut of the electrostatic potential energy along the sample x-axis. A hole quantum dotis formed at the potential maxima. The black horizontal dashed line indicates the Fermi energy, and the ellipsebelow each schematic represents single hole probability density, which is displaced and elongated as VG4 finelytunes the local electrostatic environment. d, The measured effective g-factor for a magnetic field applied alongthe sample y-axis [110]. The dashed red line is a guide to the eye.

first hole confined in a planar silicon MOSquantum dot45–48. By operating the device inthe single charge (N = 1) regime, we characterisethe g-tensor in a known and reproducibleorbital state. This allows direct comparisonsbetween experimental results and theoreticalmodeling, while the simple planar geometryallows the contributions of competing effects suchas orbital alignment and anisotropic electrode-induced strain to be separated49,50.

Our results show that electrode-inducedstrain is key in mediating electric g-factor controlin these hole MOS quantum dots. The effectof anisotropic electrode-induced strain has notbeen previously considered for hole spin qubits.Therefore, these results open a new platformfor hole spin-qubit technology, where the hole-spin qubits can be electrically manipulated bydisplacing the wave-function relative to preciselyengineered local strain gradients.

Isolating a single hole

The device studied in this work was fabricatedon an isotopically enriched 28Si substrate with ahigh-quality 5.9 nm SiO2 gate oxide. The deviceconsists of a planar multi-layer aluminum gatestack45,51. Figure 1a shows a SEM image of thedevice layout. This layout allows the formation

of a stable single hole quantum dot in the regionindicated by the white circle48. The top gate ofthe adjacent charge sensor is indicated in greenin Figure 1a. By independently monitoring thecurrent through the charge sensor (Isens), we canunambiguously identify the absolute number ofholes occupying the quantum dot. Further detailsare provided in the methods section.

Figure 1b presents the charge stabilitydiagram of the device, which was obtained bymonitoring the transconductance (dIsens/dVG2)of the charge sensor. The stability diagram showsa series of charge transitions, consistent witha single quantum dot formed under G2, withthe number of holes indicated as N . Beyondthe region labeled N = 0, no further chargetransitions were observed, confirming the devicewas operating down to the last hole.

Figure 1c shows the Coulomb charging energyfor the N = 1 to N = 2 transition, measuredat different values of VG4 (see the methodssection for full details). For a fixed numberof holes, the Coulomb charging energy (EC) isinversely proportional to the size of the quantumdot confinement43. In Figure 1c, EC increasesas VG4 is made more positive, consistent withthe quantum dot becoming smaller. Therefore,in the region between the first and secondCoulomb peaks (indicated by the colored arrows

in Figure 1b), it is possible to confine a singlehole and use VG4 to finely tune the spatial extentof the hole wavefunction.

The remainder of this paper focusses onthe g-factor of the first (N = 1) holeconfined in this planar quantum dot. The holeoccupies the lowest energy orbital state, avoidingcomplications from higher quantum dot orbitals.

Electrical modulation of the N = 1 hole g-factor

We examined the g-factor of the first hole asthe shape of the wavefunction was systematicallyvaried using the bias of a nearby electrode (G4).The effective hole g-factor was extracted fromthe linear increase in the N = 1 additionenergy with magnetic field B (see supplementaryinformation). The in-plane magnetic field wasaligned with the sample y-axis (crystal axis[110]). Figure 1d presents the single holeeffective g-factor (g110) for different electrostaticconfinement profiles. The magnitude of g110can be tuned between 1.2±0.1 and 2.6±0.1with only a small change of VG4. Fromthe maximum slope of Figure 1d we obtainthe maximum electric control over the g-factoras dg/dVG4=8.1±0.2V−1 (for this specific in-plane magnetic field orientation). The observeddg/dVG for holes is six orders of magnitude largerthan dg/dVG for electrons in identical siliconMOS devices34, and is comparable to dg/dVGobserved for holes in other Group-IV quantumdots35,52. Based on the maximum dg/dVG4,typical spin qubit conditions (B ∼ 0.2 T) couldyield a lower bound for the Rabi frequency of40 MHz35,37,44.

A key result of Figure 1d is the observationof a “sweet spot” around VG4 = −0.7 V,where dgy/dVG4 = 0. Sweet spots whered←→g /dVG = 0 V−1 are important for qubitssince the coupling between the spin and electric-fluctuations in the voltage source (VG) aresuppressed53. Minimising the effects of chargenoise is critical for hole spin qubits since thecoherence time T ∗2 of hole spins in Group-IVquantum dots is primarily limited by electricalnoise13,54. Furthermore recent theoretical workshows that it is possible to engineer sweet spotswhere the dominant charge dephasing mechanismis suppressed while still allowing high speedelectrical qubit control.23.

Characterising the g-tensor

In hole systems, the coupling to a magnetic fieldis influenced by many factors, such as the 3Dwavefunction shape, local strain, and the crystalanisotropy24,25. Fully defining the magnetic

response to a magnetic field ( ~B) requires a 3× 3g-tensor with 6 free parameters. The g-tensor isdefined as

←→g = R(ϑ0, φ0, θ0)

g1 0 00 g2 00 0 g3

R−1(ϑ0, φ0, θ0)

(1)where the angles ϑ0, φ0, and θ0 define theorientation of the principal magnetic axes withrespect to the sample (x, y, z) axes, and g1, g2and g3 define the principal g-factors. The Rmatrix represents the effect of three consecutiverotations around the sample axes (as described inthe methods), where φ0 represents the rotation ofthe g-tensor into the sample x-z frame.

Experimental characterisation of the g-tensorrequires measurements of the g-factor for arange of magnetic field orientations in all threedimensions. In this work, the g-tensor of theN = 1 hole was characterised experimentallyusing a vector magnet. The magnetic fieldwas fixed at | ~B| = 1 T, and a 2π rotationin increments of π/12 around the sample x-y, x-z and y-z axes was performed. At eachmagnetic field orientation we extracted the g-factor from the linear change of the N = 1addition energy. The measurement was repeatedfor two different confinement profiles, which werecontrolled by setting VG4 = −0.9 V (Figure 2a-i)or VG4 = −0.7 V (Figure 2b-ii). The solid lines inFigure 2a-i and Figure 2b-i show the best fit of thefull data set (all 144 points) to Equation (1). Thebest fit parameters for both confinement profilesare presented in Table 1. Figures 2a-ii and 2b-iipresent a 3D visualisation of the best fit g-tensor.

For the case of VG4 = −0.9 V, the largestprincipal g-factor (g3) is 3.9±0.1, and thesmallest principal g-factor (g1) is 1.4±0.2. Theorientation of the g-tensor is distinctly tiltedwith respect to the sample axes (Figure 2a-ii),such that the principal magnetic axes are notaligned with any lithographic or crystallographicaxes of the sample. To demonstrate this tiltedorientation, Figure 2a-iii shows the measured g-factor around the sample x-z axes on polar axes.In particular we note that the g-tensor is tiltedby φ0 = 42◦ ± 2◦ in the x-z plane. Figure 2a-iv shows a schematic of the x-z g-tensor surfacewith respect to the x-z plane of the sample,highlighting that the largest g-factor occurs whenthe magnetic field is tilted by 42◦ away fromthe Si/SiO2 interface. The observation that theprincipal axes of the g-tensor are not fixed byany sample axes is the key result of the g-tensorcharacterisation in Figure 2a.

We next investigate if the orientation ofthe g-tensor principal axes can be electricallytuned. The hole wavefunction shape was changed

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Figure 2: Electrical control of the g-tensor orientation. Measurements of the hole g-tensor for VG4 = −0.9 V(a) and VG4 = −0.7 V b. i, The measured effective g-factor for magnetic field rotations around the sample x-y(black), x-z (blue) and y-z (red) axes. Solid lines are a best fit of all data to Equation (1). Angles θ and ϕdefine the orientation of the magnetic field ~B as indicated on the adjacent sphere. ii, The shaded blue surface isthe 3D g-tensor surface defined by the appropriate parameters in Table 1. For reference the experimental data(circles/triangles) and best fit (solid lines) from i are included. iii, Reproduces the x-z g-factor measurements(θ = 0) as a polar plot. The radial axis is |g| and while the angle corresponds to ϕ. The axes of symmetry ofthe g-tensor in the x-z plane are indicated by the dashed black lines. The dashed black lines indicate the tiltedaxes of symmetry of the g-tensor around the measured x-z axes. The magnitude of the tilting into the x-z planeis indicated by the blue arrow, where the tilt in the x-z plane corresponds to φ0 of Equation (1). iv, Shows thesample schematic with the x-z g-tensor surface, highlighting the titled g-tensor orientation with respect to theSi/SiO2 interface. Triangles are used for raw data in b.

by varying VG4 from -0.9 V to -0.7 V, whileat the same time making VG2 more negative.The net effect is to strengthen the electrostaticconfinement along the sample x direction. Forthe case of VG4 = −0.7 V, the maximum principalg-factor (g3) is 1.7±0.1, while the minimumprincipal g-factor (g1) is 0.3±0.2. By comparingthe 3D g-tensor surfaces in Figure 2a-ii andFigure 2b-ii it is clear that both the size andorientation of the g-tensor are sensitive to VG4.To demonstrate the observed change in g-tensororientation, Figure 2b-iii reproduces the g-factoraround the x-z axes. For VG4 = −0.7 V wehighlight that the tilting into the sample x-zplane is now φ0 = 70◦ ± 3◦, compared to φ0 =42◦ ± 2◦ VG4 = −0.9 V. The observation thatthe orientation of the principal axes is stronglyaffected by the gate bias, even for a single hole, isthe key result of the g-tensor data set presentedin Figure 2b.

Parameter VG4 = −0.9 V VG4 = −0.7 Vg1 1.4±0.2 0.3±0.2g2 2.3±0.1 1.0±0.1g3 3.9±0.1 1.7±0.1ϑ0 19◦ ± 7◦ 10◦ ± 9◦

φ0 42◦ ± 2◦ 70◦ ± 3◦

θ0 9◦ ± 3◦ −20◦ ± 6◦

Table 1: Principle g-factors for a single hole in asilicon quantum dot, measured at two differentconfinement (VG4) potentials. The values areextracted by fitting the respective data set inFigure 2 to Equation (1). See the supplementarymaterial for the fitting procedure.

Numerical simulations of the single-hole g-tensor

To explore the physical origins of the g-tensor magnitude and orientation, we performdetailed three-dimensional modelling of thedevice, including (i) the real lithographicgate stack determined from design and devicemicroscopy, (ii) strain built-up accompanyingcool-down to cryogenic temperatures, (iii) self-consistent electrostatics with holes accumulatingat the Si/SiO2 interface at experimentally appliedgate biases, and (iv) quantum mechanics of theSi complex valence band (parameterized via the6×6 Luttinger-Kohn Hamiltonian with Bir-Pikuscoupling to the lattice strain).

In systems with strong spin-orbit coupling,the confinement and spin properties areinextricably linked. Therefore we beginthe modelling of the hole g-tensor by firstinvestigating the hole confinement profile.Figure 3a shows the calculated electrostatic

confinement potential in the x-y plane. Thehole ground state probability density projectedto the same plane is overlaid as a colour mapin Figure 3a. Similarly, Figure 3b shows thehole probability density projected to the samplex-z plane. The hole ground state is mostlyHeavy-Hole (HH) in character. The verticalextent of the wave function is ∼ 7 nm, andthe diameter is ∼ 30 nm, consistent with thediameter estimated from the measured chargingenergy in Figure 1d (see supplementary materialfor details). Therefore the holes are confinedin a thin disk-like wavefunction, which is pulledtightly against the Si/SiO2 interface. The axisof strongest orbital confinement is out-of-planewith respect to the Si/SiO2 interface.

Holes confined to a 2D-like geometry willhave the primary magnetic axis aligned with theaxis of strongest confinement17,20,32. One mighttherefore expect our disk-like hole wavefunctionto have the largest g-factor for an out-of-planemagnetic field, corresponding to φ0 ≈ 0.However, the experimental results in Figure 2show that the largest g-factor is strongly tiltedaway from the axis of strongest confinement, withφ0 >40◦.

A non-zero φ0 could, in principal, becaused by a drastic rotation of the axis ofstrongest confinement, due to a complete changein the electrostatic confinement potential (seesupplementary material). However detailednumeric simulations show that no reasonablerange of gate voltages, interface steps, or surfacescharges can produce any substantial tilting ofthe out-of-plane confinement orientation. Forall reasonable configurations, the single holeconfinement in this MOS device is most stronglydefined by the vertical hard wall potential ofthe Si/SiO2 interface. Something other thanelectrostatic confinement is therefore needed toexplain the non-zero φ0, and for this we turn tothe impact of electrode-induced strain.

Strain develops in silicon MOS devicescooled to cryogenic temperatures49,50 due todifferences in the thermal contraction betweenmetal electrodes and the silicon substrate. Inparticular, uniaxial strain alters the valenceband Heavy-Hole-Light-Hole (HH-LH) splitting∆HH-LH, while shear strains directly mix HH andLH components. Both can have an enormousinfluence on the composition of the confinedhole state and its spin properties. Figure 3cshows the spatial profile of ∆HH-LH in the activeregion of the device. The HH-LH splittingvaries by over 50% across the device, andfollows the lithography of the aluminum gatestack. The strain varies most rapidly at theedges and corners of the metal gates; shear

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Figure 3: Numcerical simulations of the electrostatic confinement, strain, and single holeeigenstates. a, Electrostatic confinement profile (contour lines) and hole ground state probability density (colourmap) in the x-y plane. The contours are spaced by 20 meV. The in-plane confinement is strongest along thesample y-axis [110], consistent with the strong influence of the large C-gate. Confinement along the sample x-axis(crystal [110] axis) is weaker, and is provided by gates G1 and G3. This simulation is for the experimentallyapplied voltages, as described in the methods. b, Hole probability density in the sample x-z plane. The footprintof each gate is shown (the vertical height of the Al gates is 30+ nm). c, Spatial profile of the strain-inducedHH-LH splitting in energy, with the hole probability density overlaid as a red colour map and HH-LH splittingshown as contours spaced by 2 meV.

strains concentrate there as well. Under thegates, biaxial compression by the shrinking metalpushes the LH basis states deeper in energyrelative to HH states, i.e., acting in the samedirection as the out-of-plane confinement. Theimpact of electrode-induced strain is particularlystrong in these silicon MOS devices due to thethin gate oxide.

In an ideal device, the hole lies directlybelow the centre of the G2 gate, as shown inFigure 3. Figure 4a presents the simulated g-tensor surface of the single hole in its groundstate (shaded blue surface). In this configurationthe g-tensor is as expected for the predominantlyHH-like state — the largest g-factor occursfor a nearly out-of-plane magnetic field, withsmall but nonzero transverse components andtilt φ0 of 8◦ due to the LH admixture. To tiltthe hole g-tensor away from the 2D plane it

is necessary to displace the hole wavefunctionaway from this point of near symmetry. Thisdisplacement can be due to atomic steps, surfacecharges, or other fluctuations of the Si/SiO2

interface. A wavefunction displacement is alsorealizable experimentally by altering the differentgate biases, such as VG4 Figure 4c presentsthe g-tensor surface simulated when the holewavefunction is electrostatically displaced byabout 10nm to a region of highly non-uniformstrain, as indicated in Figure 4d. The g-tensorsurface in Figure 4c is clearly tilted away fromthe Si/SiO2 interface with φ0 > 26◦.

Figure 4e shows the extracted tilt angle φ0of the g-tensor as the hole is artificially beingforced to various points along the sample x-axis. The spatial dependence on φ0 tracksthe strain profile, responsible also for ∆HH-LH,shown in Figure 4f. When Bir-Pikus strain

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Figure 4: Tilted g-tensor and electrode-induced strain. a, Simulated g-tensor surface for thewavefunction position and strain profile indicated in panel b. The hole is localised directly below theG2 gate, where the local strain gradients are low and shear strains are minimal. The orientation of theg-tensor is primarily defined by the orientation of the confinement, with the largest g-factor occurring forapproximately out-of-plane magnetic field (more details in e). The intersection of the g-tensor surfacewith the sample x-y (black), x-z (blue) and y-z (red) planes is indicated by the respective solid lines.c, Same as a, except forcing hole to the position indicated in d. The contour lines in b and d arespaced by 2 meV. e, Shows the g-tensor tilting (φ0) as a function of the artificial shift of the electrostaticconfinement along the sample x-axis. Red circles indicate the x-axis position and φ0 value for the casein a and c. Above the figure we schematically indicate the location of gates G1, G2, and G3. We findthat the regions near gate edges correspond to the regions of largest tilting φ0. f, Shows the simulatedstrain-indiced ∆HH-LH for a line cut along the x-axis. The ∆HH-LH is not symmetric along sample x-axis,since the lithography is not symmetric along the sample x-axis.

terms were omitted from the numerical modelthe spatial variation in the g-tensor tilting (φ0)was much reduced. These simulations suggestthat it is the non-uniform strain profile thatcauses the observed orientation of the g-tensor tobe misaligned from the electrostatic confinementorientation.

Other plausible mechanisms, such as HH-LH mixing by the microscopically low-symmetrySi/SiO2 interface55, were evaluated and deemedinsufficiently strong to significantly renormalizethe hole state g-tensor (see Supplementarymaterial for more details).

Conclusions

In this work we have experimentally studied the3D g-tensor of a single hole in a silicon MOSbased quantum dot. We characterised the full3D g-tensor for two different bias configurations.Our results demonstrate strong electric controlover both the magnitude and orientation of thesingle hole g-tensor.

A key experimental result is the wide rangeof control over the g-factor, particularly theability to configure a "sweet spot" where mostcomponents of dg/dV approach zero. For spinqubits, the coupling between hole spins andelectric fields is a balancing act, where some largecomponent of dg/dV maximises the EDSR Rabifrequency. However, a large dg/dV also amplifiesthe impact of electrical noise leading to a shorterT ∗2 . In the device under study we show that overa small range of bias configurations the hole canbe tuned in-situ to a region of high dg/dV , whichis ideal for rapid spin manipulation, then to asweet spot where dg/dV = 0 for a dominant gate,which is ideal for long lifetime qubit storage, priorto qubit readout.

Since these results are for a single hole, in thelowest quantum dot orbital state, it is possibleto compare the experimental data with detailedtheoretical models. These models suggest thatelectrode-induced strain has a significant effecton hole-spin states in p-SiMOS based quantumdots, and show how the effects of strain vary

dramatically as the hole wavefunction is movedaround with gate biases.

Electrode-induced strain has not previouslybeen considered as a mechanism enabling allelectric spin manipulation of hole-based qubits.While overall strain has been used in spin-qubitdevices56,57, it has typically been used to engineerthe static isotropic spin properties, particularlythe HH-LH splitting. This work points to apotentially new technology for spin-qubits, wherespecific gate geometries are designed to engineernon-uniform, anisotropic strain for optimisedspeed and performance.

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AcknowledgmentsThis work was funded by the AustralianResearch Council (DP150100237, DP200100147, andFL190100167) and the US Army Research Office(W911NF-17-1-0198). Devices were made at theNSW node of the Australian National FabricationFacility. D.S.M. acknowledges the support by theGeorg H. Endress foundation. K.M.I. acknowledgessupport from a Grant-in-Aid for Scientific Researchby MEXT. T.D.L. acknowledges support from theGordon Godfrey Bequest Sabbatical grant.

Author ContributionsThe devices were fabricated by F.E.H and M.V..K.M.I. prepared and supplied the 28Si wafer. S.D.Land F.M performed experiments and analysed theresults using software written by I.D.T. Device

modelling and theoretical analysis was performed byD.S.M, A.A.K, O.P.S and T.D.L. The manuscript wasprepared by S.D.L with input from all co-authors.

MethodsSample Details. The device studied in thiswork was fabricated using the same processingprocedure, but in a different processing run, asprevious planar silicon hole quantum dots devices48.During operation the R-gate is negatively biased toaccumulate a 2D hole gas at the Si/SiO2 interfacebelow. A single quantum dot is defined by positivelybiasing gates G1, G3, G4, and the C-gate. G2acts as the dot plunger gate and is operated in thenegatively biased regime. If G4 is made sufficientlynegative the device forms a double dot. Full voltagesare provided in the supplementary information. Thecharge sensor is operated by negatively biasing thesensor top gate to facilitate hole transport. Sensorbarrier gates, which are fabricated underneath thesensor top gate, are positively biased to form a regionof high dIsens/dV (either a quantum dot or sharppinch-off) which is used to charge sense the quantumdot below G2. We confirmed that the device operatesdown to a single hole using tunnel rate independentmeasurements. Further, we have confirmed thatthe orbital structure of the device studied here isconsistent with a 2D-disk-like wave function.Measurement Details. All measurements wereperformed in a BlueFors XLD dilution refrigeratorwith a base temperature of 20 mK. For chargesensor measurements we monitor dIsens/dVG2 usingthe standard dual lock-in technique with dynamicfeedback to optimise the charge sensor signal58.The charging energy and the g-factors presented inFigure 1 and Figure 2 were extracted from the spacingin VG2 between the N = 1 and N = 2 Coulombpeaks43. The spacing in VG2 was then convertedinto energy using the lever arm, ∆E = αG2∆VG2,where αG2 = (0.166±0.007) eV/V. The full data setis presented in the supplementary information. Wehave confirmed that the lever arm is independent ofthe gate voltages within the operating range of theexperiment. For all rotation measurements we firstconfirm that ∆E(B) is linear in B up to 1 T for all

directions.For the rotation matrices in Equation (1) we use thedefinition

R(ϑ0, φ0, θ0) = Rz(θ0)Ry(φ0)Rz(ϑ0) (2)

where Ry and Rz are the standard 3D rotationmatrices around y and z axes.Multiscale device modelling including strain.For modeling the hole states, we use a customnumerical framework for the construction andmultiscale simulations of the three-dimensionalmultilayer and multimaterial device model. Layoutconstruction from production masks, with attentionto their orientation with respect to the principal axesof the silicon substrate, is augmented by TEM data,process models, and known details of fabricationsteps. To account for the stress which builds upwhen cooling the heterogeneous system to cryogenictemperatures, we solve the stationary stress-strainproblem for the entire layout, assuming the deviceis unstrained at the end of fabrication and that thedevice is free of any cracks or voids.

The obtained strain pattern can be combinedwith self-consistent Schrödinger-Poisson calculations,using a Thomas–Fermi approximation to model thepartitioned two-dimensional hole gas accumulatedoutside the quantum region. The gate potentials forSchrödinger-Poisson are taken from those employedin the operation regime, including a global offset.Next, the single quantized hole in a Si complexvalence band is treated quantum-mechanically bysolving, in 3 dimensions, the 6×6 Luttinger-KohnHamiltonian with Bir-Pikus strain terms, subject torealistic electrostatic confinement and strain.

In addition, modelling of the single holestate has been conducted for a hypothetical 3Dharmonic confinement. The hypothetical modellingis illuminating due to its much simplified parameterspace of only three confinement strengths and threeangles. We extract the g-tensor by evaluatingsplittings of the ground (or excited) state doubletby the magnetic field at various angles relative to thesimulated device. Further details are presented inthe supplementary materials.